Does anyone know if there is a website that let's you search UK/European train travel - by date rather than destination?
Like skyscaner does with flights?
Not really - you mean "trains to anywhere in Switzerland" or something like that?
Remember if you are going to a big city you can search on e.g. "Paris (all stations)" - there's no need to know the specific station.
For almost all long distance trains you get the cheapest fares by booking well in advance - generally 3 months, but sometimes more. Last minute bargains are not really a thing.
There is also a specific TA Train Travel forum which this question might be better asked in.Edited: 11 May 2018, 19:00
In the UK you can look for advance fares which offer good prices. Easily searched on TOC websites. Most released 12 weeks in advance but some 24 weeks.